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Invariant Theory

Mara D. Neusel
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mthematical Library 36
BLL Rating: 

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fabio Mainardi
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If you are an undergraduate, or first-year graduate student, and you love algebra, certainly you will enjoy this book, and you will learn a lot from it. It is pleasant reading, and it is self-contained.

It begins with 30 pages of recollections on groups, rings and algebras, introducing everything will be used in the sequel. Each time a tool from commutative algebra is needed (e.g. modules, Krull dimension, integral extensions), it is carefully explained and many examples are provided. In fact, the considerable amount of examples is one of the most precious features of this book. Some examples serve as leitmotifs, since they are reconsidered and developed throughout the text, giving a concrete support to the flow of definitions and theorems. Exercises are included at the end of each chapter, making the book an ideal textbook (for an advanced undergraduate course). In addition, six sections are dedicated to the discussion of some applications of the theory to other fields, for instance to coding theory or combinatorics.

Invariant theory studies the actions of groups on algebras. For example, the group of permutations on n elements acts on the algebra of polynomials in n variables by permuting the variables, so symmetric polynomials, such as x1 + x2 + … + xn and x1x2…xn, are invariant under this action. The set of all the invariants forms a sub-algebra and it is the main object of study of the text. In the example, the algebra of invariants is generated by the elementary symmetric polynomials. For a general (finite) group acting on an algebra, it is still true that the invariants form a finitely generated algebra; this is in fact the first fundamental result in the theory. We may ask some other interesting questions: how many generators are needed in general? How can we find them? What are the relations, if any, between the generators? These, and other questions are answered in the book.

Some of the topics in algebra that are strictly related to invariant theory are:

  • linear representations of groups,
  • Coxeter groups,
  • algebraic geometry and moduli spaces.

You will certainly meet invariant theory again if you plan to specialize in any of these fields. And this book provides a friendly, yet rigorous, introduction to the subject.

I would like to stress the fact that this book is not only about a special topic in algebra, but it is also an opportunity to see many important theorems at work, whereas they are often explained in a too abstract style. Indeed, you will see concrete applications of milestones such as Hilbert’s Nullstellensatz, Noether’s normalization, and trace formulas. These are among the most beautiful achievements in algebra.

In addition, to my personal taste, the volumes from the AMS Student Mathematical Library have a very nice small format.

I strongly recommend this book for an advanced undergraduate or first-year graduate course, and also for independent study.

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at